Variational Principles in Multifractal Analysis

Part of the Progress in Mathematics book series (PM, volume 272)


Following the general concept of multifractal analysis introduced in Section 7.1, one can consider several multifractal spectra. In particular, we showed in Chapters 6 and 7 that the spectra \( \mathcal{D}_D \) and \( \mathcal{E}_E \) are analytic in several contexts. Furthermore, they coincide with the Legendre transform of certain functions defined in terms of the topological pressure, and this allows us to show that they are always convex (in fact, in certain sense, they are “generically” strictly convex; see Theorem 8.4.2). A priori it is unclear whether it is possible to effect a similar analysis in the case of the mixed multifractal spectra \( \mathcal{D}_D \) and \( \mathcal{E}_E \) which combine local and global characteristics of distinct nature. This is precisely the main theme of this chapter. In particular, we show that the mixed spectra are analytic in several contexts. The analyticity follows from a conditional variational principle for the u-dimension which is also established in this chapter, and which is important in its own right. On the other hand, we show that there are many nonconvex mixed spectra.


Variational Principle Equilibrium Measure Multifractal Analysis Multifractal Spectrum Ergodic Measure 
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© Birkhäuser Verlag AG 2008

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